{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Programming your solver\n",
    "\n",
    "In this notebook, we will look at some of the more advanced capabilities Firedrake has for configuring and developing preconditioners. In particular, we will show support for geometric multigrid, as well as user-defined preconditioners.\n",
    "\n",
    "As our prototypical example, we will consider the Stokes equations. Find $(u, p) \\in V \\times Q \\subset (H^1)^d \\times L^2$ such that\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "  \\nu\\int_\\Omega \\nabla u : \\nabla v\\,\\mathrm{d}x - \\int_\\Omega p\n",
    "  \\nabla \\cdot v\\,\\mathrm{d}x\n",
    "  &= \\int_\\Omega f \\cdot v\\,\\mathrm{d}x, \\\\\n",
    "  -\\int_\\Omega \\nabla \\cdot u q \\,\\mathrm{d}x&= 0.\n",
    "\\end{align}\n",
    "$$\n",
    "for all $(v, q) \\in V \\times Q$. Where $\\nu$ is the viscosity.\n",
    "\n",
    "We will use the inf-sup stable Taylor-Hood element pair of piecewise quadratic velocities and piecewise linear pressures."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from firedrake import *\n",
    "mesh = UnitSquareMesh(8, 8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now build a hierarchy of regularly refined meshes with this as the coarsest mesh, and grab the finest one to define the problem."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "meshes = MeshHierarchy(mesh, refinement_levels=3)\n",
    "# Grab the finest mesh\n",
    "mesh = meshes[-1]\n",
    "\n",
    "V = VectorFunctionSpace(mesh, \"CG\", 2)\n",
    "Q = FunctionSpace(mesh, \"CG\", 1)\n",
    "W = V*Q"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We set up the problem in residual form (using `TestFunction`s but no `TrialFunction`s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "v, q = TestFunctions(W)\n",
    "w = Function(W)\n",
    "u, p = split(w)\n",
    "\n",
    "nu = Constant(0.0001)\n",
    "F = nu*inner(grad(u), grad(v))*dx - p*div(v)*dx - div(u)*q*dx"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now need to augment the problem with a forcing term and boundary conditions.  We will solve a regularised lid-driven cavity problem, and thus choose $f = 0$ and the boundary conditions:\n",
    "$$\n",
    "\\begin{align}\n",
    "u &= \\begin{pmatrix}\\frac{x^2 (2 - x)^2 y^2}{4} \\\\ 0 \\end{pmatrix} & \\text{ on $\\Gamma_1 = \\{y = 1\\}$}\\\\\n",
    "u &= 0 & \\text{ otherwise}\\\\\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y = SpatialCoordinate(mesh)\n",
    "bc_value = as_vector([0.25 * x**2 * (2-x)**2 *y**2, 0])\n",
    "\n",
    "bcs = [DirichletBC(W.sub(0), bc_value, 4),\n",
    "       DirichletBC(W.sub(0), zero(mesh.geometric_dimension()), (1, 2, 3))]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This problem has a null space of constant pressures, so we'll need to inform the solver about that too."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "nullspace = MixedVectorSpaceBasis(W, [W.sub(0), VectorSpaceBasis(constant=True)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since we're going to look at a bunch of different solver options, let's have a function that builds a solver with the provided options."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def create_solver(solver_parameters, *, pmat=None, appctx=None):\n",
    "    p = {}\n",
    "    if solver_parameters is not None:\n",
    "        p.update(solver_parameters)\n",
    "    # Default to linear SNES\n",
    "    p.setdefault(\"snes_type\", \"ksponly\")\n",
    "    p.setdefault(\"ksp_rtol\", 1e-7)\n",
    "    problem = NonlinearVariationalProblem(F, w, bcs=bcs, Jp=pmat)\n",
    "    solver = NonlinearVariationalSolver(problem, nullspace=nullspace, options_prefix=\"\", \n",
    "                                        solver_parameters=p, appctx=appctx)\n",
    "    return solver"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, let's go ahead and solve the problem using a direct solver. The solver is configured with a dictionary of PETSc options. Here we select MUMPS to perform the sparse LU factorisation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "solver_parameters = {\n",
    "    \"mat_type\": \"aij\",\n",
    "    \"ksp_type\": \"preonly\",\n",
    "    # Use MUMPS since it handles the null space\n",
    "    \"pc_type\": \"lu\",\n",
    "    \"pc_factor_mat_solver_type\": \"mumps\"\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Programmatically inspect convergence of solver\n",
    "def convergence(solver):\n",
    "    from firedrake.solving_utils import KSPReasons, SNESReasons\n",
    "    snes = solver.snes\n",
    "    print(\"\"\"\n",
    "SNES iterations: {snes}; SNES converged reason: {snesreason}\n",
    "   KSP iterations: {ksp}; KSP converged reason: {kspreason}\"\"\".format(snes=snes.getIterationNumber(),\n",
    "                                                                      snesreason=SNESReasons[snes.getConvergedReason()],\n",
    "                                                                      ksp=snes.ksp.getIterationNumber(),\n",
    "                                                                      kspreason=KSPReasons[snes.ksp.getConvergedReason()]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We're ready to solve."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "SNES iterations: 1; SNES converged reason: CONVERGED_ITS\n",
      "   KSP iterations: 1; KSP converged reason: CONVERGED_ITS\n"
     ]
    }
   ],
   "source": [
    "w.assign(0)\n",
    "solver = create_solver(solver_parameters)\n",
    "solver.solve()\n",
    "convergence(solver)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now have a look at the solution, using some simple builtin plotting that utilises matplotlib."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/javascript": [
       "/* Put everything inside the global mpl namespace */\n",
       "window.mpl = {};\n",
       "\n",
       "\n",
       "mpl.get_websocket_type = function() {\n",
       "    if (typeof(WebSocket) !== 'undefined') {\n",
       "        return WebSocket;\n",
       "    } else if (typeof(MozWebSocket) !== 'undefined') {\n",
       "        return MozWebSocket;\n",
       "    } else {\n",
       "        alert('Your browser does not have WebSocket support. ' +\n",
       "              'Please try Chrome, Safari or Firefox ≥ 6. ' +\n",
       "              'Firefox 4 and 5 are also supported but you ' +\n",
       "              'have to enable WebSockets in about:config.');\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n",
       "    this.id = figure_id;\n",
       "\n",
       "    this.ws = websocket;\n",
       "\n",
       "    this.supports_binary = (this.ws.binaryType != undefined);\n",
       "\n",
       "    if (!this.supports_binary) {\n",
       "        var warnings = document.getElementById(\"mpl-warnings\");\n",
       "        if (warnings) {\n",
       "            warnings.style.display = 'block';\n",
       "            warnings.textContent = (\n",
       "                \"This browser does not support binary websocket messages. \" +\n",
       "                    \"Performance may be slow.\");\n",
       "        }\n",
       "    }\n",
       "\n",
       "    this.imageObj = new Image();\n",
       "\n",
       "    this.context = undefined;\n",
       "    this.message = undefined;\n",
       "    this.canvas = undefined;\n",
       "    this.rubberband_canvas = undefined;\n",
       "    this.rubberband_context = undefined;\n",
       "    this.format_dropdown = undefined;\n",
       "\n",
       "    this.image_mode = 'full';\n",
       "\n",
       "    this.root = $('<div/>');\n",
       "    this._root_extra_style(this.root)\n",
       "    this.root.attr('style', 'display: inline-block');\n",
       "\n",
       "    $(parent_element).append(this.root);\n",
       "\n",
       "    this._init_header(this);\n",
       "    this._init_canvas(this);\n",
       "    this._init_toolbar(this);\n",
       "\n",
       "    var fig = this;\n",
       "\n",
       "    this.waiting = false;\n",
       "\n",
       "    this.ws.onopen =  function () {\n",
       "            fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n",
       "            fig.send_message(\"send_image_mode\", {});\n",
       "            if (mpl.ratio != 1) {\n",
       "                fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n",
       "            }\n",
       "            fig.send_message(\"refresh\", {});\n",
       "        }\n",
       "\n",
       "    this.imageObj.onload = function() {\n",
       "            if (fig.image_mode == 'full') {\n",
       "                // Full images could contain transparency (where diff images\n",
       "                // almost always do), so we need to clear the canvas so that\n",
       "                // there is no ghosting.\n",
       "                fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n",
       "            }\n",
       "            fig.context.drawImage(fig.imageObj, 0, 0);\n",
       "        };\n",
       "\n",
       "    this.imageObj.onunload = function() {\n",
       "        fig.ws.close();\n",
       "    }\n",
       "\n",
       "    this.ws.onmessage = this._make_on_message_function(this);\n",
       "\n",
       "    this.ondownload = ondownload;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_header = function() {\n",
       "    var titlebar = $(\n",
       "        '<div class=\"ui-dialog-titlebar ui-widget-header ui-corner-all ' +\n",
       "        'ui-helper-clearfix\"/>');\n",
       "    var titletext = $(\n",
       "        '<div class=\"ui-dialog-title\" style=\"width: 100%; ' +\n",
       "        'text-align: center; padding: 3px;\"/>');\n",
       "    titlebar.append(titletext)\n",
       "    this.root.append(titlebar);\n",
       "    this.header = titletext[0];\n",
       "}\n",
       "\n",
       "\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n",
       "\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_canvas = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var canvas_div = $('<div/>');\n",
       "\n",
       "    canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n",
       "\n",
       "    function canvas_keyboard_event(event) {\n",
       "        return fig.key_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    canvas_div.keydown('key_press', canvas_keyboard_event);\n",
       "    canvas_div.keyup('key_release', canvas_keyboard_event);\n",
       "    this.canvas_div = canvas_div\n",
       "    this._canvas_extra_style(canvas_div)\n",
       "    this.root.append(canvas_div);\n",
       "\n",
       "    var canvas = $('<canvas/>');\n",
       "    canvas.addClass('mpl-canvas');\n",
       "    canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n",
       "\n",
       "    this.canvas = canvas[0];\n",
       "    this.context = canvas[0].getContext(\"2d\");\n",
       "\n",
       "    var backingStore = this.context.backingStorePixelRatio ||\n",
       "\tthis.context.webkitBackingStorePixelRatio ||\n",
       "\tthis.context.mozBackingStorePixelRatio ||\n",
       "\tthis.context.msBackingStorePixelRatio ||\n",
       "\tthis.context.oBackingStorePixelRatio ||\n",
       "\tthis.context.backingStorePixelRatio || 1;\n",
       "\n",
       "    mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n",
       "\n",
       "    var rubberband = $('<canvas/>');\n",
       "    rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n",
       "\n",
       "    var pass_mouse_events = true;\n",
       "\n",
       "    canvas_div.resizable({\n",
       "        start: function(event, ui) {\n",
       "            pass_mouse_events = false;\n",
       "        },\n",
       "        resize: function(event, ui) {\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
       "        },\n",
       "        stop: function(event, ui) {\n",
       "            pass_mouse_events = true;\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
       "        },\n",
       "    });\n",
       "\n",
       "    function mouse_event_fn(event) {\n",
       "        if (pass_mouse_events)\n",
       "            return fig.mouse_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    rubberband.mousedown('button_press', mouse_event_fn);\n",
       "    rubberband.mouseup('button_release', mouse_event_fn);\n",
       "    // Throttle sequential mouse events to 1 every 20ms.\n",
       "    rubberband.mousemove('motion_notify', mouse_event_fn);\n",
       "\n",
       "    rubberband.mouseenter('figure_enter', mouse_event_fn);\n",
       "    rubberband.mouseleave('figure_leave', mouse_event_fn);\n",
       "\n",
       "    canvas_div.on(\"wheel\", function (event) {\n",
       "        event = event.originalEvent;\n",
       "        event['data'] = 'scroll'\n",
       "        if (event.deltaY < 0) {\n",
       "            event.step = 1;\n",
       "        } else {\n",
       "            event.step = -1;\n",
       "        }\n",
       "        mouse_event_fn(event);\n",
       "    });\n",
       "\n",
       "    canvas_div.append(canvas);\n",
       "    canvas_div.append(rubberband);\n",
       "\n",
       "    this.rubberband = rubberband;\n",
       "    this.rubberband_canvas = rubberband[0];\n",
       "    this.rubberband_context = rubberband[0].getContext(\"2d\");\n",
       "    this.rubberband_context.strokeStyle = \"#000000\";\n",
       "\n",
       "    this._resize_canvas = function(width, height) {\n",
       "        // Keep the size of the canvas, canvas container, and rubber band\n",
       "        // canvas in synch.\n",
       "        canvas_div.css('width', width)\n",
       "        canvas_div.css('height', height)\n",
       "\n",
       "        canvas.attr('width', width * mpl.ratio);\n",
       "        canvas.attr('height', height * mpl.ratio);\n",
       "        canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n",
       "\n",
       "        rubberband.attr('width', width);\n",
       "        rubberband.attr('height', height);\n",
       "    }\n",
       "\n",
       "    // Set the figure to an initial 600x600px, this will subsequently be updated\n",
       "    // upon first draw.\n",
       "    this._resize_canvas(600, 600);\n",
       "\n",
       "    // Disable right mouse context menu.\n",
       "    $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n",
       "        return false;\n",
       "    });\n",
       "\n",
       "    function set_focus () {\n",
       "        canvas.focus();\n",
       "        canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    window.setTimeout(set_focus, 100);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items) {\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) {\n",
       "            // put a spacer in here.\n",
       "            continue;\n",
       "        }\n",
       "        var button = $('<button/>');\n",
       "        button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n",
       "                        'ui-button-icon-only');\n",
       "        button.attr('role', 'button');\n",
       "        button.attr('aria-disabled', 'false');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "\n",
       "        var icon_img = $('<span/>');\n",
       "        icon_img.addClass('ui-button-icon-primary ui-icon');\n",
       "        icon_img.addClass(image);\n",
       "        icon_img.addClass('ui-corner-all');\n",
       "\n",
       "        var tooltip_span = $('<span/>');\n",
       "        tooltip_span.addClass('ui-button-text');\n",
       "        tooltip_span.html(tooltip);\n",
       "\n",
       "        button.append(icon_img);\n",
       "        button.append(tooltip_span);\n",
       "\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    var fmt_picker_span = $('<span/>');\n",
       "\n",
       "    var fmt_picker = $('<select/>');\n",
       "    fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n",
       "    fmt_picker_span.append(fmt_picker);\n",
       "    nav_element.append(fmt_picker_span);\n",
       "    this.format_dropdown = fmt_picker[0];\n",
       "\n",
       "    for (var ind in mpl.extensions) {\n",
       "        var fmt = mpl.extensions[ind];\n",
       "        var option = $(\n",
       "            '<option/>', {selected: fmt === mpl.default_extension}).html(fmt);\n",
       "        fmt_picker.append(option);\n",
       "    }\n",
       "\n",
       "    // Add hover states to the ui-buttons\n",
       "    $( \".ui-button\" ).hover(\n",
       "        function() { $(this).addClass(\"ui-state-hover\");},\n",
       "        function() { $(this).removeClass(\"ui-state-hover\");}\n",
       "    );\n",
       "\n",
       "    var status_bar = $('<span class=\"mpl-message\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n",
       "    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
       "    // which will in turn request a refresh of the image.\n",
       "    this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_message = function(type, properties) {\n",
       "    properties['type'] = type;\n",
       "    properties['figure_id'] = this.id;\n",
       "    this.ws.send(JSON.stringify(properties));\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_draw_message = function() {\n",
       "    if (!this.waiting) {\n",
       "        this.waiting = true;\n",
       "        this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n",
       "    }\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    var format_dropdown = fig.format_dropdown;\n",
       "    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
       "    fig.ondownload(fig, format);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_resize = function(fig, msg) {\n",
       "    var size = msg['size'];\n",
       "    if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n",
       "        fig._resize_canvas(size[0], size[1]);\n",
       "        fig.send_message(\"refresh\", {});\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n",
       "    var x0 = msg['x0'] / mpl.ratio;\n",
       "    var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n",
       "    var x1 = msg['x1'] / mpl.ratio;\n",
       "    var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n",
       "    x0 = Math.floor(x0) + 0.5;\n",
       "    y0 = Math.floor(y0) + 0.5;\n",
       "    x1 = Math.floor(x1) + 0.5;\n",
       "    y1 = Math.floor(y1) + 0.5;\n",
       "    var min_x = Math.min(x0, x1);\n",
       "    var min_y = Math.min(y0, y1);\n",
       "    var width = Math.abs(x1 - x0);\n",
       "    var height = Math.abs(y1 - y0);\n",
       "\n",
       "    fig.rubberband_context.clearRect(\n",
       "        0, 0, fig.canvas.width / mpl.ratio, fig.canvas.height / mpl.ratio);\n",
       "\n",
       "    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n",
       "    // Updates the figure title.\n",
       "    fig.header.textContent = msg['label'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n",
       "    var cursor = msg['cursor'];\n",
       "    switch(cursor)\n",
       "    {\n",
       "    case 0:\n",
       "        cursor = 'pointer';\n",
       "        break;\n",
       "    case 1:\n",
       "        cursor = 'default';\n",
       "        break;\n",
       "    case 2:\n",
       "        cursor = 'crosshair';\n",
       "        break;\n",
       "    case 3:\n",
       "        cursor = 'move';\n",
       "        break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_message = function(fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function(fig, msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message(\"ack\", {});\n",
       "}\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function(fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            /* FIXME: We get \"Resource interpreted as Image but\n",
       "             * transferred with MIME type text/plain:\" errors on\n",
       "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "             * to be part of the websocket stream */\n",
       "            evt.data.type = \"image/png\";\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src);\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                evt.data);\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "        else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig[\"handle_\" + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "}\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function(e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e)\n",
       "        e = window.event;\n",
       "    if (e.target)\n",
       "        targ = e.target;\n",
       "    else if (e.srcElement)\n",
       "        targ = e.srcElement;\n",
       "    if (targ.nodeType == 3) // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "\n",
       "    // jQuery normalizes the pageX and pageY\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    // offset() returns the position of the element relative to the document\n",
       "    var x = e.pageX - $(targ).offset().left;\n",
       "    var y = e.pageY - $(targ).offset().top;\n",
       "\n",
       "    return {\"x\": x, \"y\": y};\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys (original) {\n",
       "  return Object.keys(original).reduce(function (obj, key) {\n",
       "    if (typeof original[key] !== 'object')\n",
       "        obj[key] = original[key]\n",
       "    return obj;\n",
       "  }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function(event, name) {\n",
       "    var canvas_pos = mpl.findpos(event)\n",
       "\n",
       "    if (name === 'button_press')\n",
       "    {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * mpl.ratio;\n",
       "    var y = canvas_pos.y * mpl.ratio;\n",
       "\n",
       "    this.send_message(name, {x: x, y: y, button: event.button,\n",
       "                             step: event.step,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.key_event = function(event, name) {\n",
       "\n",
       "    // Prevent repeat events\n",
       "    if (name == 'key_press')\n",
       "    {\n",
       "        if (event.which === this._key)\n",
       "            return;\n",
       "        else\n",
       "            this._key = event.which;\n",
       "    }\n",
       "    if (name == 'key_release')\n",
       "        this._key = null;\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which != 17)\n",
       "        value += \"ctrl+\";\n",
       "    if (event.altKey && event.which != 18)\n",
       "        value += \"alt+\";\n",
       "    if (event.shiftKey && event.which != 16)\n",
       "        value += \"shift+\";\n",
       "\n",
       "    value += 'k';\n",
       "    value += event.which.toString();\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, {key: value,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
       "    if (name == 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message(\"toolbar_button\", {name: name});\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n",
       "\n",
       "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.close = function() {\n",
       "        comm.close()\n",
       "    };\n",
       "    ws.send = function(m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function(msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
       "        ws.onmessage(msg['content']['data'])\n",
       "    });\n",
       "    return ws;\n",
       "}\n",
       "\n",
       "mpl.mpl_figure_comm = function(comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = $(\"#\" + id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm)\n",
       "\n",
       "    function ondownload(figure, format) {\n",
       "        window.open(figure.imageObj.src);\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy,\n",
       "                           ondownload,\n",
       "                           element.get(0));\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element.get(0);\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error(\"Failed to find cell for figure\", id, fig);\n",
       "        return;\n",
       "    }\n",
       "\n",
       "    var output_index = fig.cell_info[2]\n",
       "    var cell = fig.cell_info[0];\n",
       "\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function(fig, msg) {\n",
       "    var width = fig.canvas.width/mpl.ratio\n",
       "    fig.root.unbind('remove')\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable()\n",
       "    $(fig.parent_element).html('<img src=\"' + dataURL + '\" width=\"' + width + '\">');\n",
       "    fig.close_ws(fig, msg);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.close_ws = function(fig, msg){\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var width = this.canvas.width/mpl.ratio\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message(\"ack\", {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () { fig.push_to_output() }, 1000);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items){\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) { continue; };\n",
       "\n",
       "        var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
       "    var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
       "    button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
       "    button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
       "    buttongrp.append(button);\n",
       "    var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
       "    titlebar.prepend(buttongrp);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(el){\n",
       "    var fig = this\n",
       "    el.on(\"remove\", function(){\n",
       "\tfig.close_ws(fig, {});\n",
       "    });\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(el){\n",
       "    // this is important to make the div 'focusable\n",
       "    el.attr('tabindex', 0)\n",
       "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
       "    // off when our div gets focus\n",
       "\n",
       "    // location in version 3\n",
       "    if (IPython.notebook.keyboard_manager) {\n",
       "        IPython.notebook.keyboard_manager.register_events(el);\n",
       "    }\n",
       "    else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\n",
       "    }\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    var manager = IPython.notebook.keyboard_manager;\n",
       "    if (!manager)\n",
       "        manager = IPython.keyboard_manager;\n",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which == 13) {\n",
       "        this.canvas_div.blur();\n",
       "        event.shiftKey = false;\n",
       "        // Send a \"J\" for go to next cell\n",
       "        event.which = 74;\n",
       "        event.keyCode = 74;\n",
       "        manager.command_mode();\n",
       "        manager.handle_keydown(event);\n",
       "    }\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    fig.ondownload(fig, null);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.find_output_cell = function(html_output) {\n",
       "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
       "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
       "    // IPython event is triggered only after the cells have been serialised, which for\n",
       "    // our purposes (turning an active figure into a static one), is too late.\n",
       "    var cells = IPython.notebook.get_cells();\n",
       "    var ncells = cells.length;\n",
       "    for (var i=0; i<ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code'){\n",
       "            for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
       "                    data = data.data;\n",
       "                }\n",
       "                if (data['text/html'] == html_output) {\n",
       "                    return [cell, data, j];\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "// Register the function which deals with the matplotlib target/channel.\n",
       "// The kernel may be null if the page has been refreshed.\n",
       "if (IPython.notebook.kernel != null) {\n",
       "    IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
       "}\n"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<div id='3ad995ac-2621-43a8-aeb6-7166527d6263'></div>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib notebook\n",
    "import matplotlib.pyplot as plt\n",
    "u_h, p_h = w.split()\n",
    "fig, axes = plt.subplots()\n",
    "arrows = quiver(u_h, axes=axes)\n",
    "fig.colorbar(arrows);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Configuring a better preconditioner\n",
    "\n",
    "For this small problem, we can (and probably should) use a direct factorisation method. But what if the problem is too big? Then we need an iterative method, and an appropriate preconditioner.\n",
    "\n",
    "Let's try everyone's favourite, ILU(0)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "solver_parameters = {\n",
    "    \"mat_type\": \"aij\",\n",
    "    \"ksp_type\": \"gmres\",\n",
    "    \"ksp_gmres_modifiedgramschmidt\": None,\n",
    "    \"ksp_max_it\": 2000,\n",
    "    \"ksp_monitor\": None,\n",
    "    \"pc_type\": \"ilu\"\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "SNES iterations: 1; SNES converged reason: CONVERGED_ITS\n",
      "   KSP iterations: 1618; KSP converged reason: CONVERGED_RTOL\n"
     ]
    }
   ],
   "source": [
    "w.assign(0)\n",
    "solver = create_solver(solver_parameters)\n",
    "solver.solve()\n",
    "convergence(solver)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is, unsurprisingly, bad. Fortunately, better options are available."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Block preconditioning\n",
    "\n",
    "Firedrake hooks up all the necessary machinery to access PETSc's [`PCFIELDSPLIT`](https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/PC/PCFIELDSPLIT.html#PCFIELDSPLIT) preconditioner. This provides mechanisms for building preconditioners based on block factorisations. The Stokes problem \n",
    "$$\n",
    "\\begin{align}\n",
    "  \\nu\\int_\\Omega \\color{#800020}{\\nabla u : \\nabla v}\\,\\mathrm{d}x - \\int_\\Omega\n",
    "  \\color{#2A52BE}{p \\nabla \\cdot v}\\,\\mathrm{d}x\n",
    "  &= \\int_\\Omega f \\cdot v\\,\\mathrm{d}x, \\\\\n",
    "  -\\int_\\Omega \\color{#2A52BE}{\\nabla \\cdot u q} \\,\\mathrm{d}x&= 0.\n",
    "\\end{align}\n",
    "$$\n",
    "Is a block system\n",
    "$$\n",
    "\\mathcal{A} = \\begin{bmatrix} \\color{#800020}{A} & \\color{#2A52BE}{B^T} \\\\ \\color{#2A52BE}{B} & 0 \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "admitting a factorisation\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix} I & 0 \\\\ \\color{#2A52BE}{B} \\color{#800020}{A}^{-1} & I\\end{bmatrix}\n",
    "\\begin{bmatrix}\\color{#800020}{A} & 0 \\\\ 0 & S\\end{bmatrix}\n",
    "\\begin{bmatrix} I & \\color{#800020}{A}^{-1} \\color{#2A52BE}{B^T} \\\\ 0 & I\\end{bmatrix},\n",
    "$$\n",
    "\n",
    "with $S = -\\color{#2A52BE}{B} \\color{#800020}{A}^{-1} \\color{#2A52BE}{B^T}$ the *Schur complement*.  This has an inverse\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix} I & -\\color{#800020}{A}^{-1}\\color{#2A52BE}{B^T} \\\\ 0 & I \\end{bmatrix}\n",
    "\\begin{bmatrix} \\color{#800020}{A}^{-1} & 0 \\\\ 0 & S^{-1}\\end{bmatrix}\n",
    "\\begin{bmatrix} I & 0 \\\\ -\\color{#2A52BE}{B}\\color{#800020}{A}^{-1} & I\\end{bmatrix}.\n",
    "$$\n",
    "\n",
    "$S$ is never formed, so it's inverse is approximated using an iterative method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "exact_inverse_parameters = {\n",
    "    \"ksp_type\": \"fgmres\",\n",
    "    \"pc_type\": \"fieldsplit\",\n",
    "    \"pc_fieldsplit_type\": \"schur\",\n",
    "    \"fieldsplit_0\": {\n",
    "        \"ksp_type\": \"preonly\",\n",
    "        \"pc_type\": \"lu\",\n",
    "    },\n",
    "    \"fieldsplit_1\": {\n",
    "        \"ksp_type\": \"cg\",\n",
    "        \"ksp_rtol\": 1e-8,\n",
    "        \"pc_type\": \"none\",\n",
    "    }\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "SNES iterations: 1; SNES converged reason: CONVERGED_ITS\n",
      "   KSP iterations: 1; KSP converged reason: CONVERGED_RTOL\n"
     ]
    }
   ],
   "source": [
    "w.assign(0)\n",
    "solver = create_solver(exact_inverse_parameters)\n",
    "solver.solve()\n",
    "convergence(solver)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This looks good, but we had to use an unpreconditioned Krylov method to invert $S$. To do better we need to provide either an approximation to $S$ or $S^{-1}$.\n",
    "\n",
    "For the Stokes equations, [Silvester and Wathen (1993)](https://epubs.siam.org/doi/10.1137/0730031) show that $S \\approx -\\nu^{-1} Q$ is a good approximation, where $Q$ is the pressure mass matrix.\n",
    "\n",
    "Problem: $Q$ is not available as one of the blocks of $\\mathcal{A}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "PETSc's approach is to allow us to supply a _separate_ matrix to the solver which will be used to construct the preconditioner. So, we just need to additionally supply\n",
    "\n",
    "$$\n",
    "\\mathcal{P} = \\mathcal{A} + \\begin{bmatrix} 0 & 0 \\\\ 0 & -\\nu^{-1}Q\\end{bmatrix} = \\begin{bmatrix} \\color{#800020}{A} & \\color{#2A52BE}{B^T} \\\\ \\color{#2A52BE}{B} & -\\nu^{-1} Q \\end{bmatrix},\n",
    "$$\n",
    "where $Q = \\int_\\Omega p q \\,\\mathrm{d}x$\n",
    "\n",
    "We will construct P by symbolically computing the derivative of the residual to get $\\mathcal{A}$ and then subtracting $\\nu^{-1} Q$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "w_t = TrialFunction(W)\n",
    "_, p_t = split(w_t)\n",
    "\n",
    "pmat = lhs(derivative(F, w, w_t)) - 1/nu * p_t * q*dx"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now pass this pmat form to `create_solver` and can configure an appropriate preconditioner."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "pmat_parameters = {\n",
    "    \"mat_type\": \"nest\", # We only need the blocks\n",
    "    \"snes_type\": \"ksponly\",\n",
    "    \"ksp_view\": None,\n",
    "    \"ksp_monitor_true_residual\": None,\n",
    "    \"ksp_max_it\": 100,\n",
    "    \"pc_type\": \"fieldsplit\",\n",
    "    \"pc_fieldsplit_type\": \"schur\",\n",
    "    \"fieldsplit_0\": {\n",
    "        \"ksp_type\": \"preonly\",\n",
    "        \"pc_type\": \"lu\",\n",
    "    },\n",
    "    \"fieldsplit_1\": {\n",
    "        \"ksp_type\": \"preonly\",\n",
    "        \"pc_type\": \"lu\",\n",
    "    }\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "SNES iterations: 1; SNES converged reason: CONVERGED_ITS\n",
      "   KSP iterations: 16; KSP converged reason: CONVERGED_RTOL\n"
     ]
    }
   ],
   "source": [
    "w.assign(0)\n",
    "solver = create_solver(pmat_parameters, pmat=pmat)\n",
    "solver.solve()\n",
    "convergence(solver)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Providing auxiliary operators\n",
    "\n",
    "An inconvenience here is that we must build $\\mathcal{P}$, even though we only need $-\\nu^{-1} Q$ in additional to $\\mathcal{A}$ in the preconditioner.\n",
    "\n",
    "Firedrake offers a facilities to build Python preconditioning objects, utilising petsc4py.\n",
    "\n",
    "In this case, we can subclass the \n",
    "[`AuxiliaryOperatorPC`](https://www.firedrakeproject.org/firedrake.preconditioners.html#firedrake.preconditioners.assembled.AuxiliaryOperatorPC) to provide the mass matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "class MassMatrix(AuxiliaryOperatorPC):\n",
    "    _prefix = \"mass_\"\n",
    "    def form(self, pc, test, trial):\n",
    "        # Grab the definition of nu from the user application context (a dict)\n",
    "        nu = self.get_appctx(pc)[\"nu\"]\n",
    "        return (-1/nu * test*trial*dx, None)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we just need to select parameters such that this Python preconditioner is used."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "mass_parameters = {\n",
    "    \"mat_type\": \"nest\", # We only need the blocks\n",
    "    \"ksp_view\": None,\n",
    "    \"pc_type\": \"fieldsplit\",\n",
    "    \"pc_fieldsplit_type\": \"schur\",\n",
    "    \"fieldsplit_0\": {\n",
    "        \"ksp_type\": \"preonly\",\n",
    "        \"pc_type\": \"lu\",\n",
    "    },\n",
    "    \"fieldsplit_1\": {\n",
    "        \"ksp_type\": \"preonly\",\n",
    "        \"pc_type\": \"python\",\n",
    "        \"pc_python_type\": \"__main__.MassMatrix\",\n",
    "        \"mass_pc_type\": \"lu\",\n",
    "    }\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "SNES iterations: 1; SNES converged reason: CONVERGED_ITS\n",
      "   KSP iterations: 16; KSP converged reason: CONVERGED_RTOL\n"
     ]
    }
   ],
   "source": [
    "appctx = {\"nu\": nu} # arbitrary user data that is available inside the user PC object\n",
    "w.assign(0)\n",
    "solver = create_solver(mass_parameters, appctx=appctx)\n",
    "solver.solve()\n",
    "convergence(solver)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This performs identically to the previous approach, except that the preconditioning matrix is only built for the pressure space, and constructed \"on demand\"."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Multigrid preconditioners and smoothers\n",
    "\n",
    "So far, we've only used direct solvers for the blocks. We can also use iterative methods. Here we'll use geometric multigrid to solve\n",
    "\n",
    "In the same way that Firedrake hooks up solvers such that [`PCFIELDSPLIT`](https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/PC/PCFIELDSPLIT.html#PCFIELDSPLIT) is enabled, if a problem was defined on a mesh from a `MeshHierarchy`, [`PCMG`](https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/PC/PCMG.html) and [`SNESFAS`](https://www.mcs.anl.gov/petsc/petsc-current/docs/manualpages/SNES/SNESFASType.html) are also available."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "fieldsplit_mg_parameters = {\n",
    "    \"mat_type\": \"nest\",\n",
    "    \"ksp_view\": None,\n",
    "    \"pc_type\": \"fieldsplit\",\n",
    "    \"pc_fieldsplit_type\": \"schur\",\n",
    "    \"fieldsplit_0\": {\n",
    "        \"ksp_type\": \"preonly\",\n",
    "        \"pc_type\": \"mg\",\n",
    "        \"mg_levels\": {\n",
    "            \"ksp_type\": \"chebyshev\",\n",
    "            \"ksp_max_it\": 2,\n",
    "        }\n",
    "    },\n",
    "    \"fieldsplit_1\": {\n",
    "        \"ksp_type\": \"chebyshev\",\n",
    "        \"ksp_max_it\": 2,\n",
    "        \"pc_type\": \"python\",\n",
    "        \"pc_python_type\": \"__main__.MassMatrix\",\n",
    "        \"mass_pc_type\": \"sor\",\n",
    "    }\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, when the solver runs, PETSc will call back in to Firedrake for restriction and prolongation, as well as rediscretising $A$ on the coarser levels."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "SNES iterations: 1; SNES converged reason: CONVERGED_ITS\n",
      "   KSP iterations: 10; KSP converged reason: CONVERGED_RTOL\n"
     ]
    }
   ],
   "source": [
    "appctx = {\"nu\": nu} # arbitrary user data that is available inside the user PC object\n",
    "w.assign(0)\n",
    "solver = create_solver(fieldsplit_mg_parameters, appctx=appctx)\n",
    "solver.solve()\n",
    "convergence(solver)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also do monolithic, or \"all at once\" multigrid. Here we're using Vanka smoothing. This is supported by a new preconditioner in PETSc `PCPATCH`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "vanka_parameters = {\n",
    "    \"mat_type\": \"matfree\", # We only need the action\n",
    "    \"ksp_type\": \"fgmres\",\n",
    "    \"ksp_max_it\": 25,\n",
    "    \"pc_type\": \"mg\",\n",
    "    \"mg_levels\": {\n",
    "        \"ksp_type\": \"chebyshev\",\n",
    "        \"ksp_convergence_test\": \"skip\",\n",
    "        \"ksp_max_it\": 2,\n",
    "        \"pc_type\": \"python\",\n",
    "        \"pc_python_type\": \"firedrake.PatchPC\",\n",
    "        \"patch\": {\n",
    "            \"pc_patch_save_operators\": 1,\n",
    "            \"pc_patch_partition_of_unity\": False,\n",
    "            \"pc_patch_construct_dim\": 0,\n",
    "            # Topological decomposition\n",
    "            \"pc_patch_construct_type\": \"vanka\",\n",
    "            # Pressure space is constraint space\n",
    "            \"pc_patch_exclude_subspaces\": 1,\n",
    "            # Configure the solver on each patch\n",
    "            \"pc_patch_sub\": {\n",
    "                \"mat_type\": \"dense\",\n",
    "                \"ksp_type\": \"preonly\",\n",
    "                \"pc_type\": \"lu\",\n",
    "                \"pc_factor_shift_type\": \"nonzero\",\n",
    "            }\n",
    "        }\n",
    "    },\n",
    "    \"mg_coarse\": {\n",
    "        \"ksp_type\": \"preonly\",\n",
    "        \"pc_type\": \"python\",\n",
    "        \"pc_python_type\": \"firedrake.AssembledPC\",\n",
    "        \"assembled\": {\n",
    "            \"pc_type\": \"lu\",\n",
    "            \"pc_factor_mat_solver_type\": \"mumps\",\n",
    "        }\n",
    "    }\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "SNES iterations: 1; SNES converged reason: CONVERGED_ITS\n",
      "   KSP iterations: 12; KSP converged reason: CONVERGED_RTOL\n"
     ]
    }
   ],
   "source": [
    "w.assign(0)\n",
    "solver = create_solver(vanka_parameters)\n",
    "solver.solve()\n",
    "convergence(solver)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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